Series Solutions - Theory & Concepts

Series Solutions

Learning Objectives

Understand how to represent solutions to differential equations as power series.
Determine the radius of convergence using the Ratio Test.
Classify ordinary and singular points for variable-coefficient DEs.
Apply the method of power series around an ordinary point.
Apply the Frobenius method around a regular singular point.

When a differential equation cannot be solved by elementary analytical methods (especially when it involves variable coefficients like $x^2 y'' + y = 0$ ), we often assume the solution can be represented as an infinite power series.

Power Series Solution

We assume a solution of the form of an infinite series. Typically, we expand the series around $x_0 = 0$ (a Maclaurin series).

Power Series Solution Form

The general assumed form of a power series solution.

y(x) = \sum_{n=0}^{\infty} c_n (x-x_0)^n

Variables

Symbol	Description	Unit
$c_n$	The constant coefficients of the series	-
$x_0$	The point around which the series is expanded	-

Maclaurin Series Solution Form

The assumed form of a power series expanded around $x_0 = 0$ .

y(x) = \sum_{n=0}^{\infty} c_n x^n = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \dots

Variables

Symbol	Description	Unit
$c_n$	The constant coefficients of the series	-

Convergence and the Ratio Test

A power series is only useful if it converges to a finite value. The interval of $x$ values for which the series converges is called the interval of convergence, centered at $x_0$ , and half its length is the radius of convergence, $R$ .

Finding the Radius of Convergence

The primary tool to find $R$ is the Ratio Test. A power series $\sum a_n$ converges absolutely if the limit of the absolute ratio of successive terms is less than 1.

Applying this to the power series $a_n = c_n(x-x_0)^n$ :

Set up the limit: $\lim_{n \to \infty} \left| \frac{c_{n+1}(x-x_0)^{n+1}}{c_n(x-x_0)^n} \right| = \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right| |x-x_0| < 1$
Solve for $|x-x_0|$ : $|x-x_0| < \lim_{n \to \infty} \left| \frac{c_n}{c_{n+1}} \right| = R$

If $R=0$ , it converges only at $x=x_0$ . If $R=\infty$ , it converges for all real $x$ .

Ratio Test for Power Series

Calculates the limit to determine absolute convergence of a power series.

\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L < 1

Variables

Symbol	Description	Unit
$a_n$	The n-th term of the series	-
$L$	The limit value	-

Power Series Convergence

Before diving into solving DEs with series, it helps to visualize how a power series converges to a target function as more terms are added within its radius of convergence. This is the foundation of Taylor series approximations.

Interactive Simulation

Use the simulation below to explore how power series solutions converge as terms are added.

Power Series Solution

Solving $y' = y$ with $y(0)=1$ around the ordinary point $x_0=0$ .

y(x) \approx \sum_{n=0}^{0} \frac{x^n}{n!}

Notice how adding more terms increases the interval where the polynomial approximation closely matches the exact exponential solution.

Number of Terms (

N

LinearDegree 9

Exact Solution

e^x

Series Approximation

x ∈ [-3, 3]

Ordinary vs. Singular Points

Before solving a second-order linear DE, $P(x)y'' + Q(x)y' + R(x)y = 0$ , we must check if the expansion point $x_0$ is ordinary or singular.

Classification of Points

First, rewrite the DE in standard form: $y'' + p(x)y' + q(x)y = 0$ , where $p(x) = Q(x)/P(x)$ and $q(x) = R(x)/P(x)$ .

Ordinary Point: If both $p(x)$ and $q(x)$ are analytic (have convergent Taylor series) at $x_0$ . Practically, this means $P(x_0) \neq 0$ . Use the standard power series method.
Singular Point: If $P(x_0) = 0$ (meaning $p(x)$ or $q(x)$ is undefined at $x_0$ ).
- Regular Singular Point: If the singularities are "mild." Specifically, if $(x-x_0)p(x)$ and $(x-x_0)^2q(x)$ are both analytic at $x_0$ . Use the Frobenius Method.
- Irregular Singular Point: If the conditions for a regular singular point fail. Solutions behave wildly, and these methods generally do not apply.

Method of Power Series (Around an Ordinary Point)

STEP-BY-STEP

Assume Solution: Write $y = \sum_{n=0}^\infty c_n x^n$ .
Differentiate: Find $y' = \sum_{n=1}^\infty n c_n x^{n-1}$ and $y'' = \sum_{n=2}^\infty n(n-1) c_n x^{n-2}$ .
Substitute: Plug $y, y', y''$ into the DE.
Shift Indices: Adjust the summation indices ( $n \to k$ ) so all terms involve the exact same power of $x$ (e.g., $x^k$ ) and start at the same lower limit.
Find Recurrence Relation: Combine the sums. By the identity property of series, the coefficient of every power $x^k$ must equal zero. This gives an algebraic equation for $c_{k+2}$ in terms of $c_k$ and $c_{k-1}$ , etc.
Solve: Use the recurrence relation to find $c_2, c_3, c_4\dots$ in terms of arbitrary constants $c_0$ and $c_1$ (which correspond to initial conditions).

Method of Frobenius (Around a Regular Singular Point)

If $x_0 = 0$ is a regular singular point, standard power series fail. Instead, the solution takes the form of a Frobenius series, which adds an arbitrary power $r$ , where $c_0 \neq 0$ and $r$ is a constant to be determined (the indicial root).

Frobenius Series Solution

The assumed solution form around a regular singular point.

y(x) = \sum_{n=0}^{\infty} c_n x^{n+r} = x^r \sum_{n=0}^{\infty} c_n x^n

Variables

Symbol	Description	Unit
$c_n$	The constant coefficients of the series	-
$r$	The indicial root	-

Applying the Method of Frobenius

STEP-BY-STEP

Assume Solution: Write $y = \sum_{n=0}^{\infty} c_n x^{n+r}$ .
Find Derivatives: Differentiate twice: $y' = \sum_{n=0}^{\infty} (n+r) c_n x^{n+r-1}$ and $y'' = \sum_{n=0}^{\infty} (n+r)(n+r-1) c_n x^{n+r-2}$ .
Substitute into DE: Substitute into the differential equation, distribute any $x$ coefficients, and shift indices to align the powers of $x$ .
Indicial Equation: Look at the term with the lowest power of $x$ (usually $x^{r-2}$ or $x^r$ ) and set its coefficient to zero. Since we assumed $c_0 \neq 0$ , this yields a quadratic equation in $r$ , known as the indicial equation.
Solve for r: Find the roots $r_1, r_2$ (indicial roots). The form of the second linearly independent solution depends heavily on whether $r_1 - r_2$ is an integer or zero.

Key Takeaways

Power Series Method: Assume $y = \sum c_n x^n$ to solve variable-coefficient DEs around an Ordinary Point ( $P(x_0) \neq 0$ ).
Radius of Convergence ( $R$ ): Use the Ratio test to determine the interval $|x - x_0| < R$ where the infinite series approximation is mathematically valid.
Recurrence Relation: The core result of the method. It defines how to calculate higher-order coefficients $c_n$ based on previous ones, usually leaving $c_0$ and $c_1$ as arbitrary constants.
Frobenius Method: Required when expanding around a Regular Singular Point ( $P(x_0) = 0$ , but singularities are mild). Assumes $y = x^r \sum c_n x^n$ and requires solving an indicial equation for $r$ .

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